Without history it sucks but you're right, he probably doesn't do that with anything other than AJ, KJ, QJ, JT, J9, 22. Plus the hands you beat, J9, is a doubtful flatting hand BB vs CO preflop. QJ+ is much a much better fit preflop. It hurts to fold trips on the turn tho, but given you're facing a committing raise for 80bb more it might be best to just get away.

With some aggressive history between you in these steal situations he might try and represent the J on the turn because his hand looks exactly like a Jx after he calls your c-bet, and you're not likely to have one because your range is much bigger than his, and you could opt to check behind a marginal Jx on the flop. Still you would need to be pretty sure about that to justify putting enough air in his range to call him down at these stakes, but against some regs i would.

Back to the flop, especially since you don't seem to have history, i would consider checking behind because you have the same problem basically, unless you're sure he's gonna peel your c-bet with lots of underpairs (which he might choose to just 3bet preflop by the way) on a flop with 2 overcards, , you will get called mostly by Qx and better Jx and end up at best checking behind on later streets for pot control anyway, at worst you'll get raised on the turn and have to fold trips.

It's a trade off, but if you're ahead he has 6 outs at best for AK (doubtful flat preflop again), most often 2 outs for the overpairs you're targeting, so it's not like you need protection, and he may call a delayed c-bet on the turn anyway.

We seem to disagree often on c-bets but i thought it was worth mentioning anyway, hope it helps someone.